Matrix product solutions of boundary driven quantum chains
Matrix product solutions of boundary driven quantum chains
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DOI：
10.1088/1751-8113/48/37/373001
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发表时间：
2015-04
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影响因子：
--
通讯作者：
T. Prosen
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文献类型：
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作者：
T. Prosen
本文综述了近年来在构造边界驱动局域相互作用量子链的非平衡定态密度算符方面的进展，其中驱动是通过连接在链端的马尔可夫耗散通道实现的.我们讨论明确的解决方案，在三个不同类别的量子链，具体而言，范式（各向异性）海森堡自旋1 / 2？>链，费米-哈伯德链，和Lai-Sutherland自旋-1链，并讨论了普遍的概念，这些解决方案的特点，如矩阵产品anananomaly和一个更结构化的步行图状态anomaly。中心主题是矩阵乘积形式的非平衡态和体哈密顿量的可积结构之间的联系，如Lax算子和杨巴克斯特方程。然而，与传统的量子逆散射方法有一个显着的区别，即解决非平衡稳态密度算符需要杨-巴克斯特代数的非么正不可约表示，这些表示通常具有无限维度。这样的建设结果在非厄米，而且往往也nondiagonalisable家庭的交换转移运营商，这反过来又导致新的守恒定律的可积散装哈密顿。例如，在各向异性海森堡模型的情况下，可以构造在自旋反转（或自旋翻转）下为奇数的准局域守恒算子，而源自正统厄米转移算子（通过对数微分）的守恒算子在自旋反转下都是偶数。
We review recent progress on constructing non-equilibrium steady state density operators of boundary driven locally interacting quantum chains, where driving is implemented via Markovian dissipation channels attached to the chain’s ends. We discuss explicit solutions in three different classes of quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg spin- 1 / 2 ?> chain, the Fermi–Hubbard chain, and the Lai–Sutherland spin-1 chain, and discuss universal concepts which characterize these solutions, such as matrix product ansatz and a more structured walking graph state ansatz. The central theme is the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang–Baxter equation. However, there is a remarkable distinction with respect to the conventional quantum inverse scattering method, namely addressing nonequilibrium steady state density operators requires non-unitary irreducible representations of Yang–Baxter algebra which are typically of infinite dimensionality. Such constructions result in non-Hermitian, and often also non-diagonalisable families of commuting transfer operators which in turn result in novel conservation laws of the integrable bulk Hamiltonians. For example, in the case of the anisotropic Heisenberg model, quasi-local conserved operators which are odd under spin reversal (or spin flip) can be constructed, whereas the conserved operators stemming from orthodox Hermitian transfer operators (via logarithmic differentiation) are all even under spin reversal.